Vector Algebra

If the volume of parallelepiped formed by the vectors $\hat{i}+\lambda\,\hat{j}+\hat{k},\hat{j}+\lambda\,\hat{k}$ and $\lambda\,\hat{i}+\hat{k}$ is minimum, then $\lambda\,$ is equal to:

In a triangle $ABC$,if $|\vec{BC}|=8,|\vec{CA}|=7,|\vec{AB}|=10$,then the projection of the vector $\vec{AB}$on $\vec{AC}$ is equal to :

Let for a triangle $ABC$ $\vec{AB}=-2\hat{i}+\hat{j}+3\hat{k}$ $\vec{CB}=\alpha\,\hat{i}+\beta\,\hat{j}+\gamma\,\hat{k}$ $\vec{CA}=4\hat{i}+3\hat{j}+\delta\,\hat{k}$ If$\delta\,>0$and the area of the…

In a triangle $ABC,$if $|\vec{BC}|=3,|\vec{CA}|=5$ and $|\vec{BA}|=7,$then the projection of the vector $\vec{BA}$ on $\vec{BC}$ is equal to

If the points $P$and $Q$are respectively the circumcenter and the orthocentre of a $\Delta\,ABC$, then $\vec{PA}+\vec{PB}+\vec{PC}$ is equal to _______

Let a vector $\alpha\,\hat{i}+\beta\,\hat{j}$ be obtained by rotating the vector $\sqrt{3}\hat{i}+\hat{j}$ by an angle $45^{\circ}$ about the origin in counterclockwise direction in the first quadrant. Then the area (in…

Let$x_{0}$ be the point of local maxima of…

A vector $\vec{a}$ has components $3p$ and $1$ with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new…

Let $\vec{a}$and $\vec{b}$be the vectors along the diagonal of a parallelogram having area $2\sqrt{2}$. Let the angle between $\vec{a}$and $\vec{b}$be acute. $\left|\vec{a}\right|=1$and…

An arc $PQ$of a circle subtends a right angle at its centre $O$. The mid point of the arc $PQ$ is $R$. If$\vec{OP}=\vec{u},\vec{OR}=\vec{v}$and$\vec{OQ}=\alpha\,\vec{u}+\beta\,\vec{v}$, then$\alpha\,,\beta\,^{2}$, are…

Let $ABCD$be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $AC$ and $BD$ respectively and $\left(\vec{AB}-\vec{BC}\right)+\left(\vec{AD}-\vec{DC}\right)=k\vec{FE}$, then $k$ is equal to

In a parallelogram $ABCD,\left|\vec{AB}\right|=a,\left|\vec{AD}\right|=b&\left|\vec{AC}\right|=c$.$\vec{DB}\cdot\,\vec{AB}$has the value: